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Eun Soo Park
Office: 33-313 Telephone: 880-7221Email: [email protected] hours: by an appointment
2019Spring
04.10.2019
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“PhaseEquilibriainMaterials”
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* Monotectic reaction:
Liquid1 ↔Liquid2+ Solid
* Syntectic reaction:Liquid1+Liquid2 ↔ α
L1+L2
αK-Zn, Na-Zn,
K-Pb, Pb-U, Ca-Cd
L2+S
L1Contents for previous class
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Syntectic reaction
Liquid1+Liquid2 ↔ α αK-Zn, Na-Zn,K-Pb, Pb-U, Ca-Cd
Contents for previous class
Positive ΔHm
Liquid1+Liquid2
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AV
BV
Barrier of Heterogeneous Nucleation
4)coscos32(
316)(
316*
3
2
3
2
3
V
SL
V
SL
GS
GG
3* 2 3cos cos
4*
sub homoG G
32 3cos cos ( )4
A
A B
V SV V
How about the nucleation at the crevice or at the edge?
* *hom( )hetG S G
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How do we treat the non-spherical shape?
* * Asub homo
A B
VG GV V
Substrate
Good Wetting
AV
BV Substrate
Bad Wetting
AV
BV
Effect of good and bad wetting on substrate
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Contents for today’s class
Chapter 6Binary Phase Diagrams: Reactions in the Solid State
* Eutectoid reaction: α ↔ β + γ
* Monotectoid reaction: α1 ↔ β + α2
* Peritectoid reaction: α + β ↔ γ
Chapter 7Binary Phase Diagram: Allotropy of the Components
* SYSTEMS IN WHICH TWO PHASES ARE IN EQUILIBRIUM WITH THE LIQUID PHASE
* SYSTEMS IN WHICH ONE PHASE IS IN EQUILIBRIUM WITH THE LIQUID PHASE
* Metatectic reaction: β ↔ L + α Ex. Co-Os, Co-Re and Co-Ru
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Eutectoid reaction: α ↔ β + γ
Monotectoid reaction: α1 ↔ β + α2
Peritectoid reaction: α + β ↔ γ
* Transformation can only proceed if –ΔGbulk > +ΔGinterface+ΔGstrain
Disordered atomic arrangement at grain boundaries will reduce the strain energy factor and the interfacial energy needed to nucleate a new phase.
The finer the grain size, and hence the larger the grain boundary area, the more readily will the transformation proceed. “allotropic transformation”
Chapter 6Binary Phase Diagrams: Reactions in the Solid State~ Only the kinetics of the reaction differ
By nucleation and growth mechanism ~ Strain energy factor~ nucleation sites of transformation
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Iron-Carbon System
Source: Reed-Hill, Abbaschian, Physical Metallurgy Principles, 3rd Edition, PWS Publishing Company, 1994.
Diagram is not ever plotted past 12 wt%
Cementite
Hägg carbide
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Iron Carbon Phase Diagram
ACM
A3
Ae1 (Eutectoid Temperature)
Formation of Ledeburite
Formation of Pearlite
Source: Reed-Hill, Abbaschian, Physical Metallurgy Principles, 3rd Edition, PWS Publishing Company, 1994.
Steel Cast Irons
FCC
ferriteBCC
ferrite, BCC
Eutectoid reaction: γ ↔ α + Fe3C
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Cementite – What is it?
• Cementite has an orthorhombic lattice with approximate parameters 0.45165, 0.50837 and 0.67297 nm.
• There are twelve iron atoms and four carbon atoms per unit cell, corresponding to the formula Fe3C.
Source: http://www.msm.cam.ac.uk/phase‐trans/2003/Lattices/cementite.htmlH. K. D. H. Bhadeshia
Purple: Carbon atomsOrange: Iron atoms
Iron Carbide – Ceramic Compound
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Pearlite: What is it?
• The eutectoid transformation:γ (0.77% C) α (0.02%C) + Fe3C (6.67%C)
• Alternate lamellae of ferrite and cementite as thecontinuous phase
• Diffusional Transformation
• “Pearlite” name is related to the regular array of thelamellae in colonies. Etching attacks the ferrite phasemore than the cementite. The raised and regularlyspaced cementite lamellae act as diffraction gratings anda pearl-like luster is produced by the diffraction of lightof various wavelengths from different colonies
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Pearlite• Twophasesappearindefiniteratiobytheleverrule:
• Sincethedensitiesaresame(7.86and7.4)lamellaewidthsare7:1
• Heterogeneousnucleationandgrowthofpearlitecolonies– buttypicallygrowsintoonly1grain
Reed-Hill, Abbaschian, 1994, [5]
%8867.6
77.067.6
%1267.6
077.0
cementite
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Lamellae Nucleation
Reed-Hill, Abbaschian, 1994
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Lamellae Nucleation
Reed-Hill, Abbaschian, 1994
Fig. Growing cementite and ferrite lamellae may nucleate each other.
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Eutectic Solidification (Kinetics)If isnucleatedfromliquidandstartstogrow,whatwouldbe
thecompositionatthe interface of/Ldetermined?
→roughinterface(diffusioninterface) &localequilibrium
Howaboutat/L? Nature’schoice?Lamellarstructure
Whatwouldbearoleofthecurvature atthetip?
→Gibbs‐ThomsonEffect
interlamellarspacing→ 1)λ ↓ → eutecticgrowthrate↑
but2)λ ↓ → α/β interfacialE,γαβ↑→ lowerlimitofλ
B‐richliquid
A‐richliquid
B‐richliquid
i i SA G minimum →G=Gbulk +Ginterface =G0 + AMisfitstrainenergyInterfaceenergy+
Eutecticsolidification:diffusioncontrolledprocess
→ fastestgrowthrateatacertainλ
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Interlamellar Spacing• Interlamellarspacingλ isalmostconstantinpearliteformedfromγ atafixedT.
• Temperaturehasastrongeffectonspacing– lowerT(largeΔT)promotessmallerλ.
– Pearliteformedat700oChasλ ~1mmandRockwellC– 15.
– Pearliteformedat600oChasλ ~0.1mmandRockwellC– 40.
• ZenerandHillertEq.forspacing(eq.4.39):
THT
V
ECFe
3/4
σa/Fe3C=Interfacialenergyperunitareaofa/Fe3CboundaryTE =Theequilibriumtemperature(Ae1)ΔHV =ThechangeinenthalpyperunitvolumeΔT =TheundercoolingbelowAe1
IH5:deriveλwithmaximumgrowthrateatafixedT(eutecticcase)
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Effect of Undercooling on
Krauss, Steels, 1995
ΔT λundercooling
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Effect of Interlamellar Spacing
Stone et al, 1975
Rc
Y.S.
λHardness
Yield Strength
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Iron-Carbon (Fe-C) Phase Diagram
3 invariant points:
-Eutectoid (B) [0.77 %C]:
+Fe3C
-Eutectic (A) [4.32 %C]:
L +Fe3C
Fe3C
(ce
men
tite
)
1600
1400
1200
1000
800
600
4000 1 2 3 4 5 6 6.7
L
(austenite)
+L
+Fe3C
+Fe3C
L+Fe3C
(Fe) Co, wt% C
1148°C
T(°C)
727°C = T eutectoid
ASR
4.30
R S
0.76
Ceu
tect
oid
B
Fe3C (cementite-hard) (ferrite-soft)
C
-Peritectic (C) [0.17%C]:
L +
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Hypoeutectoid Steel
Fe3C
(ce
men
tite
)
1600
1400
1200
1000
800
600
4000 1 2 3 4 5 6 6.7
L
(austenite)
+L
+ Fe3C
+ Fe3C
L+Fe3C
(Fe) Co , wt% C
1148°C
T (°C)
727°C
C0
0.76
R S
w =S /(R +S )
w Fe3C=(1- w )
w pearlite = w pearlite
r s
w =s /(r +s )w =(1- w )
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Proeuctectoid Ferrite – Pearlite
0.38 wt% C: Plain Carbon – Medium Carbon Steel
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6.3 Monotectoid reaction: α1 ↔ β+α2
βZr ↔ α + βTa
Both βZr and βTa have the same crystal structure (b.c.c.) but different lattice spacing.
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α’ ↔ α + β
Both α and α‘ are face-centered cubic phases, differing only in lattice spacing.
α’ ↔ α +γ
Monotectoid reaction: α1 ↔ β+α2
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Peritectoid reaction: α + β ↔ γ
Revision of Cu-Al phase diagram
This peritectoid reaction is claimed to be very sluggish, and heat treatment times up to 14 weeks were necessary to produce the reatcion.
Cu-Al phase diagram
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Peritectoid reaction: α + β ↔ γ
Ag-Al phase diagram (schematic)
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Peritectoid reaction: η+ε ↔ η’ Eutectoid reaction: η ↔ η’+ α
Order-disorder transformations
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CHAPTER 7
Binary Phase Diagrams. Allotropy of the Components
Several commercially important metals exist in more than one crystalline form.
Ex. Iron- three allotropes α, γ, δ
Titanium – two allotropes close-packed hexagonal
α Ti stable at low temp. and body-
centered cubic β Ti stable at high temp.
Plutonium – six allotropes _ the highest number
of modifications
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Such systems can be further divided according to whether the hightemperature allotrope forms a continuous series of solid solutions with theother component or not.
βZr ↔ α + βTa
Monotectoid reaction:
Eutectoidal decomposition
a. SYSTEMS IN WHICH ONE PHASE IS IN EQUILIBRIUM WITH THE LIQUID PHASE
7.1.1. The high temperature phase forms a series of solid solutions with the other component
Loop-shaped phase region
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Types of phase diagrams formed when the hightemperature allotrope forms a continuous series ofsolid solutions with the second component.
7.1.1. The high temperature phase forms a series of solid solutions with the other component
allotropic modification allotropic modification
(a) single component have two allotropic modifications.
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allotropic modification
7.1.1. The high temperature phase forms a series of solid solutions with the other component
X α α+β αT1 T2 T3
Y α β αLoop-shaped phase region
(a) single component have two allotropic modifications.
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7.1.1. The high temperature phase forms a series of solid solutions with the other component
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Eutectoidal decompositionof high temperature allotrope β into α
and γ, the low temperature allotropes of components A and B respectively.
(b) Both components have two allotropic modifications.
Complete series of solid solutions are formed between each of the allotropes
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Complete series of solid solutions are formed between each of the allotropes in the system Ti-Zr.
7.1.1. The high temperature phase forms a series of solid solutions with the other component
(b) Both components have two allotropic modifications.
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7.1.2. Both phases form limited solid solutions with the other component
High-temperature β phase, as well as the low-temperature α phase formlimited solid solutions with component B.
a. SYSTEMS IN WHICH ONE PHASE IS IN EQUILIBRIUM WITH THE LIQUID PHASE
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Simple eutectic system with solid-soild phase transitions
Polymorphism: the ability of a solid material to exist in more than one form or crystal structure
Eutectoid reaction: α ↔ β + γ Peritectoid reaction: α + β ↔ γ
(Both α and β are allotropes of A)
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b. SYSTEMS IN WHICH TWO PHASES ARE IN EQUILIBRIUM WITH THE LIQUID PHASE
Metatectic reaction: β ↔ L + α Ex. Co-Os, Co-Re and Co-Ru
L+ α & L + β or L + β & L + γ or L + γ & L + δ
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Metatectic reaction: β ↔ L + α Ex. Co-Os, Co-Re and Co-Ru(Both α and β are allotropes of A)
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Metatectic reaction: β ↔ L + α Ex. Co-Os, Co-Re and Co-Ru
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L+ α & L + β or L + β & L + γ or L + γ & L + δ
b. SYSTEMS IN WHICH TWO PHASES ARE IN EQUILIBRIUM WITH THE LIQUID PHASE
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L+ α & L + β or L + β & L + γb. SYSTEMS IN WHICH TWO PHASES ARE IN EQUILIBRIUM WITH THE LIQUID PHASE
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42Metatectic reaction: β ↔ L + α Ex. Co-Os, Co-Re, Co-Ru
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Syntectic reaction
Liquid1+Liquid2 ↔ α
L1+L2
α
K-Zn, Na-Zn,K-Pb, Pb-U, Ca-Cd
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MIDTERM: April1910‐12AM
Scopes: Text ~ page 117/ Teaching note ~10
QUIZs and Homeworks
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Eutectic Solidification (Kinetics)If isnucleatedfromliquidandstartstogrow,whatwouldbe
thecompositionatthe interface of/Ldetermined?
→roughinterface(diffusioninterface) &localequilibrium
Howaboutat/L? Nature’schoice?Lamellarstructure
Whatwouldbearoleofthecurvature atthetip?
→Gibbs‐ThomsonEffect
interlamellarspacing→ 1)λ ↓ → eutecticgrowthrate↑
but2)λ ↓ → α/β interfacialE,γαβ↑→ lowerlimitofλ
B‐richliquid
A‐richliquid
B‐richliquid
i i SA G minimum →G=Gbulk +Ginterface =G0 + AMisfitstrainenergyInterfaceenergy+
Eutecticsolidification:diffusioncontrolledprocess
→ fastestgrowthrateatacertainλ
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2( ) mVG
0( )E
H TGT
Criticalspacing, * *: ( ) 0G
*
0
2 E mT VH T
Whatwouldbetheminimum?
Foraninterlamellarspacing,,thereisatotalof(2/)m2
of/ interfaceperm3 ofeutectic(단위 부피당 계면 E).
( ) ?G
Howmany/ interfacesperunitlength? 21
2mG V
2( ) mVG
m
L TGT
Eutectic Solidification
Drivingforcefornucleation=interfacialEateutecticphase
λ →∞ ,
*
SolidificationwilltakeplaceifΔGisnegative(‐).
Molarvolume
Forverylargevaluesofλ, interfacialE~0 Totalundercooling
InterfacialEterm
46
최소 층상 간격
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*
0
2 E mT V identical to critical radiusH T
*GrowthMechanism:Gibbs‐ThomsoneffectinaG‐compositiondiagram?
All3phasesareinequilibrium.
The cause of G increase is the curvature of theα/L and β/L interfaces arising from the need tobalance the interfacial tensions at the α/β/Ltriple point, therefore the increase will bedifferent for the two phases, but for simplecases it can be shown to be for both.
*)
:
SL SL m
V V
v
2 2 T 1cf rG L T
L latent heat per unit volume
L =ΔH=HL‐ HS
in pure metal
1)Atλ=λ*(<∞),
Gibbs-Thomson effect
1) If λ=λ*, growth rate will be infinitelyslow because the liquid in contact withboth phases has the same composition,XE in Figure 4.32.
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B concentration ahead of the β phase
B concentration ahead of the α phase
<
2)Atλ=(∞>)λ (>λ*),
ConcentrationofBmustbehigheraheadoftheα phaseBrejectedfromtheα phase→thetipsofthegrowingβ
∞>λ >λ*,총 계면에너지 감소로 Gα andGβ arecorrespondinglyreduced.
Fig.4.33(a)Molarfreeenergydiagramat(TE ‐ ∆T0)forthecaseλ*<λ<∞ ,showingthecompositiondifferenceavailabletodrivediffusionthroughtheliquid(∆X).(b)Modelusedtocalculatethegrowthrate.
* Eutectic growth rate, v→ if α/L and β/L interfaces are highly mobile
→ proportional to flux of solute through liquid
→ diffusion controlled process
→XBL/α >XBL/β
)( // LB
LB XX
dldCD 1/effective diffusion distance.. 1/λ
)( // LB
LB XX
dldCD
XDk 1
0
*
,0,XX
X
00
*
0 )1(
TX
XX
)1(*
02
TDkv
Maximum growth rate at a fixed T0*2
(next page)
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49Fig. 4.34 Eutectic phase diagram showing the relationship
between ∆X and ∆X0 (exaggerated for clarity)
0
*
,0,XX
X
00
*
0 )1(
TX
XX
00
0 1(
TX
XX
과냉도가 적을 경우,
ΔX will it self depend on λ. ~ maximum value, ΔX0